Fundamental Trade-Offs in Multi-Bit Watermarking of Stochastic Processes

TL;DR

This paper explores the fundamental trade-offs in multi-bit watermarking of stochastic processes, balancing false alarms, distortion, and decoding reliability through an information-theoretic framework.

Contribution

It formulates the watermarking problem as a distributional information embedding and hypothesis testing task, deriving bounds and benchmarks for optimal trade-offs.

Findings

Derived fundamental bounds on watermarking performance metrics.

Established scheme-agnostic benchmarks for watermarking methods.

Presented a practical watermarking scheme illustrating theoretical trade-offs.

Abstract

We study multi-bit watermarking for data generated by stochastic processes, where a hidden message is embedded during sampling and must be decodable by an authorized detector that possesses side information unavailable to unauthorized observers. In high-stakes deployments, a practical watermark must simultaneously control false alarms, preserve generation quality without distorting the output distribution, and support reliable multi-bit decoding. Satisfying all three goals at once inevitably creates fundamental trade-offs. We formulate watermark embedding as a distributional information-embedding problem and watermark detection as a multiple-hypothesis testing problem under distortion and rate constraints, leading to four fundamental metrics: false-alarm probability, detection error probability, distortion, and information rate. Within this information-theoretic framework, we derive matched converse and achievability bounds that characterize the optimal trade-offs and provide scheme-agnostic benchmarks for any watermarking method. For stationary ergodic stochastic processes, we further obtain matched asymptotic limits and connect them to the finite-sample regime. Finally, we present a reference watermarking construction satisfying our assumptions and empirically illustrating the predicted trade-offs.